math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 16

Speedup: 1.0×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. re = 8.799645276538189e+231
  2. im = 6.37428425514748e-171

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

   2. cancel-sign-sub 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   3. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

   4. sub-neg 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

   5. remove-double-neg 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

   6. neg-sub0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \]
6. Add Preprocessing

Alternative 2: 64.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.9 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+150}:\\ \;\;\;\;re \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 72000000000.0)
     (sin re)
     (if (<= im 4.9e+119)
       (sqrt (/ 0.0625 (pow re 4.0)))
       (if (<= im 1.5e+150) (* re t_0) (* (sin re) t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re);
	} else if (im <= 4.9e+119) {
		tmp = sqrt((0.0625 / pow(re, 4.0)));
	} else if (im <= 1.5e+150) {
		tmp = re * t_0;
	} else {
		tmp = sin(re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im ** 2.0d0)
    if (im <= 72000000000.0d0) then
        tmp = sin(re)
    else if (im <= 4.9d+119) then
        tmp = sqrt((0.0625d0 / (re ** 4.0d0)))
    else if (im <= 1.5d+150) then
        tmp = re * t_0
    else
        tmp = sin(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 4.9e+119) {
		tmp = Math.sqrt((0.0625 / Math.pow(re, 4.0)));
	} else if (im <= 1.5e+150) {
		tmp = re * t_0;
	} else {
		tmp = Math.sin(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(im, 2.0)
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re)
	elif im <= 4.9e+119:
		tmp = math.sqrt((0.0625 / math.pow(re, 4.0)))
	elif im <= 1.5e+150:
		tmp = re * t_0
	else:
		tmp = math.sin(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (im ^ 2.0))
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 4.9e+119)
		tmp = sqrt(Float64(0.0625 / (re ^ 4.0)));
	elseif (im <= 1.5e+150)
		tmp = Float64(re * t_0);
	else
		tmp = Float64(sin(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im ^ 2.0);
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 4.9e+119)
		tmp = sqrt((0.0625 / (re ^ 4.0)));
	elseif (im <= 1.5e+150)
		tmp = re * t_0;
	else
		tmp = sin(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 72000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.9e+119], N[Sqrt[N[(0.0625 / N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.5e+150], N[(re * t$95$0), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 7.2e10 < im < 4.89999999999999996e119

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   7. Taylor expanded in re around 0 8.8%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 8.8%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

      2. sqrt-unprod 18.9%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}} \cdot \frac{0.25}{{re}^{2}}}} \]

      3. frac-times 18.9%

         \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot 0.25}{{re}^{2} \cdot {re}^{2}}}} \]

      4. metadata-eval 18.9%

         \[\leadsto \sqrt{\frac{\color{blue}{0.0625}}{{re}^{2} \cdot {re}^{2}}} \]

      5. pow-prod-up 18.9%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{re}^{\left(2 + 2\right)}}}} \]

      6. metadata-eval 18.9%

         \[\leadsto \sqrt{\frac{0.0625}{{re}^{\color{blue}{4}}}} \]

   9. Applied egg-rr 18.9%

      \[\leadsto \color{blue}{\sqrt{\frac{0.0625}{{re}^{4}}}} \]

   if 4.89999999999999996e119 < im < 1.50000000000000006e150

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 6.4%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 6.4%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 6.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 6.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 6.4%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 6.4%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 6.4%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 6.4%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   13. Taylor expanded in re around 0 3.1%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 3.1%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} \]

       2. *-commutative 3.1%

          \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   15. Simplified 3.1%

       \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   if 1.50000000000000006e150 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 95.2%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 95.2%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 95.2%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 95.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 95.2%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 95.2%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 95.2%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 95.2%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.9 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re \cdot \left(t\_0 + 1\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+150}:\\ \;\;\;\;re \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 72000000000.0)
     (* (sin re) (+ t_0 1.0))
     (if (<= im 7e+119)
       (sqrt (/ 0.0625 (pow re 4.0)))
       (if (<= im 1.16e+150) (* re t_0) (* (sin re) t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re) * (t_0 + 1.0);
	} else if (im <= 7e+119) {
		tmp = sqrt((0.0625 / pow(re, 4.0)));
	} else if (im <= 1.16e+150) {
		tmp = re * t_0;
	} else {
		tmp = sin(re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im ** 2.0d0)
    if (im <= 72000000000.0d0) then
        tmp = sin(re) * (t_0 + 1.0d0)
    else if (im <= 7d+119) then
        tmp = sqrt((0.0625d0 / (re ** 4.0d0)))
    else if (im <= 1.16d+150) then
        tmp = re * t_0
    else
        tmp = sin(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re) * (t_0 + 1.0);
	} else if (im <= 7e+119) {
		tmp = Math.sqrt((0.0625 / Math.pow(re, 4.0)));
	} else if (im <= 1.16e+150) {
		tmp = re * t_0;
	} else {
		tmp = Math.sin(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(im, 2.0)
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re) * (t_0 + 1.0)
	elif im <= 7e+119:
		tmp = math.sqrt((0.0625 / math.pow(re, 4.0)))
	elif im <= 1.16e+150:
		tmp = re * t_0
	else:
		tmp = math.sin(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (im ^ 2.0))
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = Float64(sin(re) * Float64(t_0 + 1.0));
	elseif (im <= 7e+119)
		tmp = sqrt(Float64(0.0625 / (re ^ 4.0)));
	elseif (im <= 1.16e+150)
		tmp = Float64(re * t_0);
	else
		tmp = Float64(sin(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im ^ 2.0);
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re) * (t_0 + 1.0);
	elseif (im <= 7e+119)
		tmp = sqrt((0.0625 / (re ^ 4.0)));
	elseif (im <= 1.16e+150)
		tmp = re * t_0;
	else
		tmp = sin(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 72000000000.0], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7e+119], N[Sqrt[N[(0.0625 / N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.16e+150], N[(re * t$95$0), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 79.7%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 79.7%

         \[\leadsto \color{blue}{1 \cdot \sin re} + 0.5 \cdot \left({im}^{2} \cdot \sin re\right) \]

      2. associate-*r* 79.7%

         \[\leadsto 1 \cdot \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      3. distribute-rgt-out 79.7%

         \[\leadsto \color{blue}{\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   7. Simplified 79.7%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   if 7.2e10 < im < 7.0000000000000001e119

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   7. Taylor expanded in re around 0 8.8%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 8.8%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

      2. sqrt-unprod 18.9%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}} \cdot \frac{0.25}{{re}^{2}}}} \]

      3. frac-times 18.9%

         \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot 0.25}{{re}^{2} \cdot {re}^{2}}}} \]

      4. metadata-eval 18.9%

         \[\leadsto \sqrt{\frac{\color{blue}{0.0625}}{{re}^{2} \cdot {re}^{2}}} \]

      5. pow-prod-up 18.9%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{re}^{\left(2 + 2\right)}}}} \]

      6. metadata-eval 18.9%

         \[\leadsto \sqrt{\frac{0.0625}{{re}^{\color{blue}{4}}}} \]

   9. Applied egg-rr 18.9%

      \[\leadsto \color{blue}{\sqrt{\frac{0.0625}{{re}^{4}}}} \]

   if 7.0000000000000001e119 < im < 1.16000000000000008e150

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 6.4%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 6.4%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 6.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 6.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 6.4%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 6.4%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 6.4%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 6.4%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   13. Taylor expanded in re around 0 3.1%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 3.1%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} \]

       2. *-commutative 3.1%

          \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   15. Simplified 3.1%

       \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   if 1.16000000000000008e150 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 95.2%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 95.2%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 95.2%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 95.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 95.2%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 95.2%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 95.2%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 95.2%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.15e-12)
   (sin re)
   (if (<= im 1.35e+154)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.15e-12) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.15d-12) then
        tmp = sin(re)
    else if (im <= 1.35d+154) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = sin(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.15e-12) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.15e-12:
		tmp = math.sin(re)
	elif im <= 1.35e+154:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = math.sin(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.15e-12)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.15e-12)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = sin(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.15e-12], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.14999999999999995e-12

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.5%

      \[\leadsto \color{blue}{\sin re} \]

   if 1.14999999999999995e-12 < im < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 61.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 100.0%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242}:\\ \;\;\;\;re \cdot \left(t\_0 + 1\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+254}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 72000000000.0)
     (sin re)
     (if (<= im 6.2e+119)
       (sqrt (/ 0.0625 (pow re 4.0)))
       (if (<= im 3.1e+242)
         (* re (+ t_0 1.0))
         (if (<= im 7.2e+254)
           (* 2.0 (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))
           (* re t_0)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re);
	} else if (im <= 6.2e+119) {
		tmp = sqrt((0.0625 / pow(re, 4.0)));
	} else if (im <= 3.1e+242) {
		tmp = re * (t_0 + 1.0);
	} else if (im <= 7.2e+254) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	} else {
		tmp = re * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im ** 2.0d0)
    if (im <= 72000000000.0d0) then
        tmp = sin(re)
    else if (im <= 6.2d+119) then
        tmp = sqrt((0.0625d0 / (re ** 4.0d0)))
    else if (im <= 3.1d+242) then
        tmp = re * (t_0 + 1.0d0)
    else if (im <= 7.2d+254) then
        tmp = 2.0d0 * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    else
        tmp = re * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.2e+119) {
		tmp = Math.sqrt((0.0625 / Math.pow(re, 4.0)));
	} else if (im <= 3.1e+242) {
		tmp = re * (t_0 + 1.0);
	} else if (im <= 7.2e+254) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	} else {
		tmp = re * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(im, 2.0)
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re)
	elif im <= 6.2e+119:
		tmp = math.sqrt((0.0625 / math.pow(re, 4.0)))
	elif im <= 3.1e+242:
		tmp = re * (t_0 + 1.0)
	elif im <= 7.2e+254:
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	else:
		tmp = re * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (im ^ 2.0))
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 6.2e+119)
		tmp = sqrt(Float64(0.0625 / (re ^ 4.0)));
	elseif (im <= 3.1e+242)
		tmp = Float64(re * Float64(t_0 + 1.0));
	elseif (im <= 7.2e+254)
		tmp = Float64(2.0 * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	else
		tmp = Float64(re * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im ^ 2.0);
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 6.2e+119)
		tmp = sqrt((0.0625 / (re ^ 4.0)));
	elseif (im <= 3.1e+242)
		tmp = re * (t_0 + 1.0);
	elseif (im <= 7.2e+254)
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = re * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 72000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.2e+119], N[Sqrt[N[(0.0625 / N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 3.1e+242], N[(re * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+254], N[(2.0 * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 7.2e10 < im < 6.1999999999999999e119

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   7. Taylor expanded in re around 0 8.8%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 8.8%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

      2. sqrt-unprod 18.9%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}} \cdot \frac{0.25}{{re}^{2}}}} \]

      3. frac-times 18.9%

         \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot 0.25}{{re}^{2} \cdot {re}^{2}}}} \]

      4. metadata-eval 18.9%

         \[\leadsto \sqrt{\frac{\color{blue}{0.0625}}{{re}^{2} \cdot {re}^{2}}} \]

      5. pow-prod-up 18.9%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{re}^{\left(2 + 2\right)}}}} \]

      6. metadata-eval 18.9%

         \[\leadsto \sqrt{\frac{0.0625}{{re}^{\color{blue}{4}}}} \]

   9. Applied egg-rr 18.9%

      \[\leadsto \color{blue}{\sqrt{\frac{0.0625}{{re}^{4}}}} \]

   if 6.1999999999999999e119 < im < 3.1000000000000001e242

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 78.3%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 78.3%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 78.3%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 78.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in re around 0 54.7%

       \[\leadsto \left(0.5 \cdot {im}^{2} + 1\right) \cdot \color{blue}{re} \]

   if 3.1000000000000001e242 < im < 7.19999999999999954e254

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 3.1%

         \[\leadsto \color{blue}{2 \cdot \sin re} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{2 \cdot \sin re} \]

   9. Taylor expanded in re around 0 72.3%

      \[\leadsto 2 \cdot \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \]

   if 7.19999999999999954e254 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 100.0%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   13. Taylor expanded in re around 0 81.8%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 81.8%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} \]

       2. *-commutative 81.8%

          \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   15. Simplified 81.8%

       \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+254}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242}:\\ \;\;\;\;re \cdot \left(t\_0 + 1\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+254}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 72000000000.0)
     (sin re)
     (if (<= im 4.8e+119)
       (pow re -2.0)
       (if (<= im 3.1e+242)
         (* re (+ t_0 1.0))
         (if (<= im 7.2e+254)
           (* 2.0 (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0)))))
           (* re t_0)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re);
	} else if (im <= 4.8e+119) {
		tmp = pow(re, -2.0);
	} else if (im <= 3.1e+242) {
		tmp = re * (t_0 + 1.0);
	} else if (im <= 7.2e+254) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * pow(re, 2.0))));
	} else {
		tmp = re * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im ** 2.0d0)
    if (im <= 72000000000.0d0) then
        tmp = sin(re)
    else if (im <= 4.8d+119) then
        tmp = re ** (-2.0d0)
    else if (im <= 3.1d+242) then
        tmp = re * (t_0 + 1.0d0)
    else if (im <= 7.2d+254) then
        tmp = 2.0d0 * (re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0))))
    else
        tmp = re * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 4.8e+119) {
		tmp = Math.pow(re, -2.0);
	} else if (im <= 3.1e+242) {
		tmp = re * (t_0 + 1.0);
	} else if (im <= 7.2e+254) {
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0))));
	} else {
		tmp = re * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(im, 2.0)
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re)
	elif im <= 4.8e+119:
		tmp = math.pow(re, -2.0)
	elif im <= 3.1e+242:
		tmp = re * (t_0 + 1.0)
	elif im <= 7.2e+254:
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0))))
	else:
		tmp = re * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (im ^ 2.0))
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 4.8e+119)
		tmp = re ^ -2.0;
	elseif (im <= 3.1e+242)
		tmp = Float64(re * Float64(t_0 + 1.0));
	elseif (im <= 7.2e+254)
		tmp = Float64(2.0 * Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0)))));
	else
		tmp = Float64(re * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im ^ 2.0);
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 4.8e+119)
		tmp = re ^ -2.0;
	elseif (im <= 3.1e+242)
		tmp = re * (t_0 + 1.0);
	elseif (im <= 7.2e+254)
		tmp = 2.0 * (re * (1.0 + (-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = re * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 72000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.8e+119], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 3.1e+242], N[(re * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+254], N[(2.0 * N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 7.2e10 < im < 4.8e119

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.8%

      \[\leadsto \color{blue}{\sin re} \]

   7. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 8.8%

      \[\leadsto {\color{blue}{re}}^{-2} \]

   if 4.8e119 < im < 3.1000000000000001e242

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 78.3%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 78.3%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 78.3%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 78.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in re around 0 54.7%

       \[\leadsto \left(0.5 \cdot {im}^{2} + 1\right) \cdot \color{blue}{re} \]

   if 3.1000000000000001e242 < im < 7.19999999999999954e254

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\sin re + \sin re} \]
   7. Step-by-step derivation

      1. count-2 3.1%

         \[\leadsto \color{blue}{2 \cdot \sin re} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{2 \cdot \sin re} \]

   9. Taylor expanded in re around 0 72.3%

      \[\leadsto 2 \cdot \color{blue}{\left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)} \]

   if 7.19999999999999954e254 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 100.0%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   13. Taylor expanded in re around 0 81.8%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 81.8%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} \]

       2. *-commutative 81.8%

          \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   15. Simplified 81.8%

       \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+254}:\\ \;\;\;\;2 \cdot \left(re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242} \lor \neg \left(im \leq 6.2 \cdot 10^{+254}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 72000000000.0)
   (sin re)
   (if (<= im 5e+119)
     (pow re -2.0)
     (if (or (<= im 3.1e+242) (not (<= im 6.2e+254)))
       (* re (* 0.5 (pow im 2.0)))
       (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re);
	} else if (im <= 5e+119) {
		tmp = pow(re, -2.0);
	} else if ((im <= 3.1e+242) || !(im <= 6.2e+254)) {
		tmp = re * (0.5 * pow(im, 2.0));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * pow(re, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 72000000000.0d0) then
        tmp = sin(re)
    else if (im <= 5d+119) then
        tmp = re ** (-2.0d0)
    else if ((im <= 3.1d+242) .or. (.not. (im <= 6.2d+254))) then
        tmp = re * (0.5d0 * (im ** 2.0d0))
    else
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 5e+119) {
		tmp = Math.pow(re, -2.0);
	} else if ((im <= 3.1e+242) || !(im <= 6.2e+254)) {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	} else {
		tmp = re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re)
	elif im <= 5e+119:
		tmp = math.pow(re, -2.0)
	elif (im <= 3.1e+242) or not (im <= 6.2e+254):
		tmp = re * (0.5 * math.pow(im, 2.0))
	else:
		tmp = re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 5e+119)
		tmp = re ^ -2.0;
	elseif ((im <= 3.1e+242) || !(im <= 6.2e+254))
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	else
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 5e+119)
		tmp = re ^ -2.0;
	elseif ((im <= 3.1e+242) || ~((im <= 6.2e+254)))
		tmp = re * (0.5 * (im ^ 2.0));
	else
		tmp = re * (1.0 + (-0.16666666666666666 * (re ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 72000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 5e+119], N[Power[re, -2.0], $MachinePrecision], If[Or[LessEqual[im, 3.1e+242], N[Not[LessEqual[im, 6.2e+254]], $MachinePrecision]], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 7.2e10 < im < 4.9999999999999999e119

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.8%

      \[\leadsto \color{blue}{\sin re} \]

   7. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 8.8%

      \[\leadsto {\color{blue}{re}}^{-2} \]

   if 4.9999999999999999e119 < im < 3.1000000000000001e242 or 6.2000000000000004e254 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 84.7%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 84.7%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 84.7%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 84.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 84.7%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 84.7%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 84.7%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 84.7%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   13. Taylor expanded in re around 0 62.8%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 62.8%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} \]

       2. *-commutative 62.8%

          \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   15. Simplified 62.8%

       \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   if 3.1000000000000001e242 < im < 6.2000000000000004e254

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 3.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0 72.3%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242} \lor \neg \left(im \leq 6.2 \cdot 10^{+254}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242}:\\ \;\;\;\;re \cdot \left(t\_0 + 1\right)\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+254}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 72000000000.0)
     (sin re)
     (if (<= im 6.2e+119)
       (pow re -2.0)
       (if (<= im 3.1e+242)
         (* re (+ t_0 1.0))
         (if (<= im 6.2e+254)
           (* re (+ 1.0 (* -0.16666666666666666 (pow re 2.0))))
           (* re t_0)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re);
	} else if (im <= 6.2e+119) {
		tmp = pow(re, -2.0);
	} else if (im <= 3.1e+242) {
		tmp = re * (t_0 + 1.0);
	} else if (im <= 6.2e+254) {
		tmp = re * (1.0 + (-0.16666666666666666 * pow(re, 2.0)));
	} else {
		tmp = re * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im ** 2.0d0)
    if (im <= 72000000000.0d0) then
        tmp = sin(re)
    else if (im <= 6.2d+119) then
        tmp = re ** (-2.0d0)
    else if (im <= 3.1d+242) then
        tmp = re * (t_0 + 1.0d0)
    else if (im <= 6.2d+254) then
        tmp = re * (1.0d0 + ((-0.16666666666666666d0) * (re ** 2.0d0)))
    else
        tmp = re * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(im, 2.0);
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.2e+119) {
		tmp = Math.pow(re, -2.0);
	} else if (im <= 3.1e+242) {
		tmp = re * (t_0 + 1.0);
	} else if (im <= 6.2e+254) {
		tmp = re * (1.0 + (-0.16666666666666666 * Math.pow(re, 2.0)));
	} else {
		tmp = re * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(im, 2.0)
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re)
	elif im <= 6.2e+119:
		tmp = math.pow(re, -2.0)
	elif im <= 3.1e+242:
		tmp = re * (t_0 + 1.0)
	elif im <= 6.2e+254:
		tmp = re * (1.0 + (-0.16666666666666666 * math.pow(re, 2.0)))
	else:
		tmp = re * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (im ^ 2.0))
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 6.2e+119)
		tmp = re ^ -2.0;
	elseif (im <= 3.1e+242)
		tmp = Float64(re * Float64(t_0 + 1.0));
	elseif (im <= 6.2e+254)
		tmp = Float64(re * Float64(1.0 + Float64(-0.16666666666666666 * (re ^ 2.0))));
	else
		tmp = Float64(re * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im ^ 2.0);
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 6.2e+119)
		tmp = re ^ -2.0;
	elseif (im <= 3.1e+242)
		tmp = re * (t_0 + 1.0);
	elseif (im <= 6.2e+254)
		tmp = re * (1.0 + (-0.16666666666666666 * (re ^ 2.0)));
	else
		tmp = re * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 72000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.2e+119], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 3.1e+242], N[(re * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.2e+254], N[(re * N[(1.0 + N[(-0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 7.2e10 < im < 6.1999999999999999e119

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.8%

      \[\leadsto \color{blue}{\sin re} \]

   7. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 8.8%

      \[\leadsto {\color{blue}{re}}^{-2} \]

   if 6.1999999999999999e119 < im < 3.1000000000000001e242

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 78.3%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 78.3%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 78.3%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 78.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in re around 0 54.7%

       \[\leadsto \left(0.5 \cdot {im}^{2} + 1\right) \cdot \color{blue}{re} \]

   if 3.1000000000000001e242 < im < 6.2000000000000004e254

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 3.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0 72.3%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]

   if 6.2000000000000004e254 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 100.0%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   13. Taylor expanded in re around 0 81.8%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 81.8%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} \]

       2. *-commutative 81.8%

          \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   15. Simplified 81.8%

       \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+254}:\\ \;\;\;\;re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242} \lor \neg \left(im \leq 6.2 \cdot 10^{+254}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 72000000000.0)
   (sin re)
   (if (<= im 5.5e+119)
     (pow re -2.0)
     (if (or (<= im 3.1e+242) (not (<= im 6.2e+254)))
       (* re (* 0.5 (pow im 2.0)))
       (* -0.16666666666666666 (pow re 3.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re);
	} else if (im <= 5.5e+119) {
		tmp = pow(re, -2.0);
	} else if ((im <= 3.1e+242) || !(im <= 6.2e+254)) {
		tmp = re * (0.5 * pow(im, 2.0));
	} else {
		tmp = -0.16666666666666666 * pow(re, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 72000000000.0d0) then
        tmp = sin(re)
    else if (im <= 5.5d+119) then
        tmp = re ** (-2.0d0)
    else if ((im <= 3.1d+242) .or. (.not. (im <= 6.2d+254))) then
        tmp = re * (0.5d0 * (im ** 2.0d0))
    else
        tmp = (-0.16666666666666666d0) * (re ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 5.5e+119) {
		tmp = Math.pow(re, -2.0);
	} else if ((im <= 3.1e+242) || !(im <= 6.2e+254)) {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	} else {
		tmp = -0.16666666666666666 * Math.pow(re, 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re)
	elif im <= 5.5e+119:
		tmp = math.pow(re, -2.0)
	elif (im <= 3.1e+242) or not (im <= 6.2e+254):
		tmp = re * (0.5 * math.pow(im, 2.0))
	else:
		tmp = -0.16666666666666666 * math.pow(re, 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 5.5e+119)
		tmp = re ^ -2.0;
	elseif ((im <= 3.1e+242) || !(im <= 6.2e+254))
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	else
		tmp = Float64(-0.16666666666666666 * (re ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif (im <= 5.5e+119)
		tmp = re ^ -2.0;
	elseif ((im <= 3.1e+242) || ~((im <= 6.2e+254)))
		tmp = re * (0.5 * (im ^ 2.0));
	else
		tmp = -0.16666666666666666 * (re ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 72000000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 5.5e+119], N[Power[re, -2.0], $MachinePrecision], If[Or[LessEqual[im, 3.1e+242], N[Not[LessEqual[im, 6.2e+254]], $MachinePrecision]], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 7.2e10 < im < 5.5000000000000003e119

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.8%

      \[\leadsto \color{blue}{\sin re} \]

   7. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 8.8%

      \[\leadsto {\color{blue}{re}}^{-2} \]

   if 5.5000000000000003e119 < im < 3.1000000000000001e242 or 6.2000000000000004e254 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Taylor expanded in im around 0 84.7%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 84.7%

         \[\leadsto \sin re + \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in 84.7%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Simplified 84.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 84.7%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 84.7%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \sin re} \]

       2. *-commutative 84.7%

          \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   12. Simplified 84.7%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   13. Taylor expanded in re around 0 62.8%

       \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot re\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 62.8%

          \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot re} \]

       2. *-commutative 62.8%

          \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   15. Simplified 62.8%

       \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot {im}^{2}\right)} \]

   if 3.1000000000000001e242 < im < 6.2000000000000004e254

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 3.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0 72.3%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]

   7. Taylor expanded in re around inf 72.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {re}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+119}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+242} \lor \neg \left(im \leq 6.2 \cdot 10^{+254}\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+241} \lor \neg \left(im \leq 6.2 \cdot 10^{+254}\right):\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 72000000000.0)
   (sin re)
   (if (or (<= im 2.3e+241) (not (<= im 6.2e+254)))
     (pow re -2.0)
     (* -0.16666666666666666 (pow re 3.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re);
	} else if ((im <= 2.3e+241) || !(im <= 6.2e+254)) {
		tmp = pow(re, -2.0);
	} else {
		tmp = -0.16666666666666666 * pow(re, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 72000000000.0d0) then
        tmp = sin(re)
    else if ((im <= 2.3d+241) .or. (.not. (im <= 6.2d+254))) then
        tmp = re ** (-2.0d0)
    else
        tmp = (-0.16666666666666666d0) * (re ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re);
	} else if ((im <= 2.3e+241) || !(im <= 6.2e+254)) {
		tmp = Math.pow(re, -2.0);
	} else {
		tmp = -0.16666666666666666 * Math.pow(re, 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re)
	elif (im <= 2.3e+241) or not (im <= 6.2e+254):
		tmp = math.pow(re, -2.0)
	else:
		tmp = -0.16666666666666666 * math.pow(re, 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif ((im <= 2.3e+241) || !(im <= 6.2e+254))
		tmp = re ^ -2.0;
	else
		tmp = Float64(-0.16666666666666666 * (re ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re);
	elseif ((im <= 2.3e+241) || ~((im <= 6.2e+254)))
		tmp = re ^ -2.0;
	else
		tmp = -0.16666666666666666 * (re ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 72000000000.0], N[Sin[re], $MachinePrecision], If[Or[LessEqual[im, 2.3e+241], N[Not[LessEqual[im, 6.2e+254]], $MachinePrecision]], N[Power[re, -2.0], $MachinePrecision], N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 7.2e10 < im < 2.2999999999999999e241 or 6.2000000000000004e254 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.7%

      \[\leadsto \color{blue}{\sin re} \]

   7. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 9.1%

      \[\leadsto {\color{blue}{re}}^{-2} \]

   if 2.2999999999999999e241 < im < 6.2000000000000004e254

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 3.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0 72.3%

      \[\leadsto \color{blue}{re \cdot \left(1 + -0.16666666666666666 \cdot {re}^{2}\right)} \]

   7. Taylor expanded in re around inf 72.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {re}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+241} \lor \neg \left(im \leq 6.2 \cdot 10^{+254}\right):\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 72000000000.0) (sin re) (pow re -2.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re);
	} else {
		tmp = pow(re, -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 72000000000.0d0) then
        tmp = sin(re)
    else
        tmp = re ** (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re);
	} else {
		tmp = Math.pow(re, -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re)
	else:
		tmp = math.pow(re, -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = sin(re);
	else
		tmp = re ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re);
	else
		tmp = re ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 72000000000.0], N[Sin[re], $MachinePrecision], N[Power[re, -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 7.2e10 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.7%

      \[\leadsto \color{blue}{\sin re} \]

   7. Applied egg-rr 8.5%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 8.4%

      \[\leadsto {\color{blue}{re}}^{-2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;{re}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 72000000000.0) (sin re) (* (/ 0.5 re) (/ 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = sin(re);
	} else {
		tmp = (0.5 / re) * (0.5 / re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 72000000000.0d0) then
        tmp = sin(re)
    else
        tmp = (0.5d0 / re) * (0.5d0 / re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 / re) * (0.5 / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 72000000000.0:
		tmp = math.sin(re)
	else:
		tmp = (0.5 / re) * (0.5 / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 / re) * Float64(0.5 / re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = sin(re);
	else
		tmp = (0.5 / re) * (0.5 / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 72000000000.0], N[Sin[re], $MachinePrecision], N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   if 7.2e10 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 8.5%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   7. Taylor expanded in re around 0 8.4%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 8.4%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

      2. sqrt-div 8.4%

         \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      3. metadata-eval 8.4%

         \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      4. sqrt-pow1 25.6%

         \[\leadsto \frac{0.5}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      5. metadata-eval 25.6%

         \[\leadsto \frac{0.5}{{re}^{\color{blue}{1}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      6. pow1 25.6%

         \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      7. sqrt-div 25.6%

         \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

      8. metadata-eval 25.6%

         \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

      9. sqrt-pow1 8.4%

         \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}} \]

      10. metadata-eval 8.4%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{{re}^{\color{blue}{1}}} \]

      11. pow1 8.4%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   9. Applied egg-rr 8.4%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.5% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 72000000000.0) re (* (/ 0.5 re) (/ 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = re;
	} else {
		tmp = (0.5 / re) * (0.5 / re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 72000000000.0d0) then
        tmp = re
    else
        tmp = (0.5d0 / re) * (0.5d0 / re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 72000000000.0) {
		tmp = re;
	} else {
		tmp = (0.5 / re) * (0.5 / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 72000000000.0:
		tmp = re
	else:
		tmp = (0.5 / re) * (0.5 / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 72000000000.0)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 / re) * Float64(0.5 / re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 72000000000.0)
		tmp = re;
	else
		tmp = (0.5 / re) * (0.5 / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 72000000000.0], re, N[(N[(0.5 / re), $MachinePrecision] * N[(0.5 / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.2e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 63.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0 31.5%

      \[\leadsto \color{blue}{re} \]

   if 7.2e10 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 8.5%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   7. Taylor expanded in re around 0 8.4%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 8.4%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

      2. sqrt-div 8.4%

         \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      3. metadata-eval 8.4%

         \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      4. sqrt-pow1 25.6%

         \[\leadsto \frac{0.5}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      5. metadata-eval 25.6%

         \[\leadsto \frac{0.5}{{re}^{\color{blue}{1}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      6. pow1 25.6%

         \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      7. sqrt-div 25.6%

         \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

      8. metadata-eval 25.6%

         \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

      9. sqrt-pow1 8.4%

         \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}} \]

      10. metadata-eval 8.4%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{{re}^{\color{blue}{1}}} \]

      11. pow1 8.4%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   9. Applied egg-rr 8.4%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 72000000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{re} \cdot \frac{0.5}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.8% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 1.0) re 1.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		tmp = re;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.0d0) then
        tmp = re
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.0) {
		tmp = re;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.0:
		tmp = re
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.0)
		tmp = re;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.0)
		tmp = re;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.0], re, 1.0]

Derivation

1. Split input into 2 regimes

2. if re < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 45.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0 29.4%

      \[\leadsto \color{blue}{re} \]

   if 1 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

      2. cancel-sign-sub 100.0%

         \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      3. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

      4. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

      5. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 5.3%

      \[\leadsto \color{blue}{\frac{\sin re \cdot -2}{\sin re \cdot -2 + \left(\sin re \cdot -2 - \sin re \cdot -2\right)}} \]
   7. Step-by-step derivation

      1. +-inverses 5.3%

         \[\leadsto \frac{\sin re \cdot -2}{\sin re \cdot -2 + \color{blue}{0}} \]

      2. +-rgt-identity 5.3%

         \[\leadsto \frac{\sin re \cdot -2}{\color{blue}{\sin re \cdot -2}} \]

      3. *-inverses 5.3%

         \[\leadsto \color{blue}{1} \]

   8. Simplified 5.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

   2. cancel-sign-sub 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   3. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

   4. sub-neg 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

   5. remove-double-neg 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

   6. neg-sub0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
4. Add Preprocessing

6. Applied egg-rr 2.8%

   \[\leadsto \color{blue}{\log \left({1}^{\sin re}\right)} \]
7. Step-by-step derivation

   1. pow-base-1 2.8%

      \[\leadsto \log \color{blue}{1} \]

   2. metadata-eval 2.8%

      \[\leadsto \color{blue}{0} \]

8. Simplified 2.8%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 2.8%

   \[\leadsto 0 \]
10. Add Preprocessing

Alternative 16: 4.8% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) + e^{im} \cdot \left(0.5 \cdot \sin re\right)} \]

   2. cancel-sign-sub 100.0%

      \[\leadsto \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \left(-e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   3. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - \left(-e^{im}\right)\right)} \]

   4. sub-neg 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-\left(-e^{im}\right)\right)\right)} \]

   5. remove-double-neg 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

   6. neg-sub0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
4. Add Preprocessing

6. Applied egg-rr 3.9%

   \[\leadsto \color{blue}{\frac{\sin re \cdot -2}{\sin re \cdot -2 + \left(\sin re \cdot -2 - \sin re \cdot -2\right)}} \]
7. Step-by-step derivation

   1. +-inverses 3.9%

      \[\leadsto \frac{\sin re \cdot -2}{\sin re \cdot -2 + \color{blue}{0}} \]

   2. +-rgt-identity 3.9%

      \[\leadsto \frac{\sin re \cdot -2}{\color{blue}{\sin re \cdot -2}} \]

   3. *-inverses 3.9%

      \[\leadsto \color{blue}{1} \]

8. Simplified 3.9%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 3.9%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024069 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))